Scaling laws of impact induced shock pressure and particle velocity in planetary mantle

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Scaling laws of impact induced shock pressure and particle velocity in planetary mantle

J. Monteux, J. Arkani-Hamed

To cite this version:

J. Monteux, J. Arkani-Hamed. Scaling laws of impact induced shock pressure and particle velocity in planetary mantle. Icarus, Elsevier, 2016, 264, pp.246-256. �10.1016/j.icarus.2015.09.040�. �hal- 01637422�

1

Scaling Laws of Impact Induced Shock Pressure and Particle Velocity in Planetary Mantle

1

2

J. Monteux

3

Laboratoire Magmas et Volcans, Clermont-Ferrand, France

4

5

J. Arkani-Hamed

6

Department of Physics, University of Toronto, Toronto, Canada

7

Department of Earth and Planetary Sciences, McGill University, Montreal, Canada

8

9 10 11

Abstract. While major impacting bodies during accretion of a Mars type planet have very low 12

velocities (<10 km/s), the characteristics of the shockwave propagation and, hence, the derived

13

scaling laws are poorly known for these low velocity impacts. Here, we use iSALE-2D

14

hydrocode simulations to calculate shock pressure and particle velocity in a Mars type body for

15

impact velocities ranging from 4 to 10 km/s. Large impactors of 100 to 400 km in diameter,

16

comparable to those impacted on Mars and created giant impact basins, are examined. To better

17

represent the power law distribution of shock pressure and particle velocity as functions of

18

distance from the impact site at the surface, we propose three distinct regions in the mantle: a

19

near field regime, which extends to 1-3 times the projectile radius into the target, where the peak

20

shock pressure and particle velocity decay very slowly with increasing distance, a mid field

21

region, which extends to ~ 4.5 times the impactor radius, where the pressure and particle velocity

22

decay exponentially but moderately, and a more distant far field region where the pressure and

23

particle velocity decay strongly with distance. These scaling laws are useful to determine impact

24

heating of a growing proto-planet by numerous accreting bodies.

25 26 27 28 29 30 31

2

Introduction:

32 33

Small planets are formed by accreting a huge number of planetesimals, a few km to a few tens of

34

km in size, in the solar nebula [e.g., Wetherill and Stewart, 1989; Matsui, 1993: Chambers and

35

Wetherill, 1998; Kokubo and Ida, 1995, 1996, 1998, 2000; Wetherill and Inaba, 2000; Rafikov,

36

2003; Chambers, 2004; Raymond, et al., 2006]. An accreting body may generate shock wave if

37

the impact-induced pressure in the target exceeds the elastic Hugoniot pressure, ~3 GPa,

38

implying that collision of a planetesimal with a growing planetary embryo can generate shock

39

waves when the embryo’s radius exceeds 150 km, assuming that impact occurs at the escape

40

velocity of the embryo and taking the mean density of the embryo and projectile to be 3000

41

kg/m

³

. Hundreds of thousands of collisions must have occurred during the formation of small

42

planets such as Mercury and Mars when they were orbiting the Sun inside a dense population of

43

planetesimal. Such was also the case during the formation of embryos that later were accreted to

44

produce Venus and Earth. Terrestrial planets have also experienced large high velocity impacts

45

after their formation. Over 20 giant impact basins on Mars with diameters larger than 1000 km

46

[Frey, 2008)], the Caloris basin on Mercury with a 1550 km diameter, and the South Pole Aitken

47

basin on Moon with a 2400 km diameter are likely created during catastrophic bombardment

48

period at around 4 Ga. The overlapping Rheasilvia and Veneneia basins on 4-Vesta are probably

49

created by projectiles with an impact velocity of about 5 km/s within the last 1-2 Gy [Keil et al.,

50

1997; Schenk et al., 2012].

51

52 53

The shock wave produced by an impact when the embryo is undifferentiated and completely

54

solid propagates as a spherical wave centered at the impact site until it reaches the surface of the

55

embryo in the opposite side. Each impact increases the temperature of the embryo within a

56

region near the impact site. Because impacts during accretion occur from different directions, the

57

mean temperature in the upper parts of the embryo increases almost globally. On the other hand,

58

the shock wave produced by a large impact during the heavy bombardment period must have

59

increased the temperature in the mantle and the core of the planets directly beneath the impact

60

site, enhancing mantle convection [e.g., Watters et al., 2009; Roberts and Arkani-Hamed, 2012,

61

3

2014], modifying the CMB heat flux which could in turn favour a hemispheric dynamo on Mars

62

[Monteux et al., 2015], or crippling the core dynamo [e.g., Arkani-Hamed and Olson, 2010a].

63

The impact-induced shock pressure inside a planet has been investigated by numerically solving

64

the shock dynamic equations using hydrocode simulations [e.g., Pierazzo et al., 1997;

65

Wuennemann and Ivanov, 2003; Wuennemann et al., 2006; Bar and Citron, 2011; Kraus et al.,

66

2011; Ivanov et al., 2010; Bierhaus et al., 2012] or finite difference techniques [e.g., Ahrens and

67

O’Keefe, 1987; Mitani, 2003]. However, these numerical solutions demand considerable

68

computer capacity and time and are not practical for investigating the huge number of impacts

69

that occur during the growth of a planet. Hence, the scaling laws derived from field experiments

70

[e.g., Perret and Bass, 1975; Melosh, 1989] or especially from hydrocode simulations [Pierazzo

71

et al., 1997] are of great interest when considering the full accretionary history of a planetary

72

objects [e.g. Senshu et al., 2002, Monteux et al., 2014] or when measuring the influence of a

73

single large impact on the long-term thermal evolution of deep planetary interiors [e.g. Monteux

74

et al., 2007, 2009, 2013, Ricard et al, 2009; Roberts et al., 2009; Arkani-Hamed and Olson,

75

2010a; Arkani-Hamed and Ghods, 2011]. Although the scaling laws provide approximate

76

estimates of the shock pressure distribution, their simplicity and the small differences between

77

their results and those obtained by the hydrocode simulations of the shock dynamic equations

78

(that are likely within the numerical errors that could have been introduced due to the uncertainty

79

of the physical parameters used in the hydrocode models) make them a powerful tool that can be

80

combined with other geophysical approaches such as dynamo models [e.g. Monteux, et al., 2015]

81

or convection models [e.g. Watters et al, 2009; Roberts and Arkani-Hamed, 2012, 2014].

82 83

During the decompression of shocked material much of the internal energy of the shock state is

84

converted into heat leading to a temperature increase below the impact site. The present study

85

focuses on deriving scaling laws of shock pressure and particle velocity distributions in silicate

86

mantle of a planet on the basis of several hydrocode simulations. The scaling laws of Pierazzo et

87

al. [1997] were derived using impact velocities of 10 to 80 km/s, hence may not be viable at low

88

impact velocities. For example, at an impact velocity of 5 km/s, comparable to the escape

89

velocity of Mars, the shock pressure scaling law provides an unrealistic shock pressure that

90

increases with depth. Here we model shock pressure and particle velocity distributions in the

91

mantle using hydrocode simulations for impact velocities of 4 to 10 km/s and projectile

92

4

diameters ranging from 100 to 400 km, as an attempt to extend Pierazzo et al.’s [1997] scaling

93

laws to low impact velocities and reasonable impactor radii occurring during the formation of

94

terrestrial planets. Hence, on the basis of our scaling laws it is possible to estimate the

95

temperature increase as a function of depth below the impact site for impact velocities

96

compatible with the accretionary conditions of terrestrial protoplanets. These scaling laws can

97

easily be implemented in a multi-impact approach [e.g. Senshu et al, 2003, Monteux et al., 2014]

98

to monitor the temperature evolution inside a growing protoplanet whereas it is not yet possible

99

to adopt hydrocode simulations for that purpose.

100 101

The hydrocode models we have calculated are described in the first section, while the second

102

section presents the scaling laws derived from the hydrocode models. The concluding remarks

103

are relegated to the third section.

104 105

Hydrocode models of shock pressure distribution:

106 107

The huge number of impacts during accretion makes it impractical to consider oblique impacts.

108

Not only it requires formidable computer time, but more importantly because of the lack of

109

information about the impact direction, i.e. the impact angle relative to vertical and azimuth

110

relative to north. Therefore, we consider only head-on collisions (vertical impact) to model the

111

thermo-mechanical evolution during an impact between a differentiated Mars size body and a

112

large impactor. We use the iSALE-2D axisymmetric hydrocode, which is a multi-rheology,

113

multi-material hydrocode, specifically developed to model impact crater formation on a

114

planetary scale [Collins et al., 2004, Davison et al., 2010]. To limit computation time, we use a 2

115

km spatial resolution (i.e. more than 25 cells per projectile radius, cppr) and a maximum time

116

step of 0.05 s which is sufficient to describe the shockwave propagation through the entire

117

mantle. The minimum post impact monitoring time is set to the time needed by the shockwave to

118

reach the core-mantle boundary (≈ 5 minutes for the impact velocities studied here).

119 120

We investigate the shock pressure and particle velocity distributions inside a Mars size model

121

planet for impact velocities V

imp

of 4 to 10 km/s and impactors of 100 to 400 km in diameter.

122

Such impactors are capable of creating basins of 1000 to 2500 km in diameter according to

123

5

Schmidt and Housen [1987] and Holsapple [1993] scaling relationships between the impactor

124

diameter and the resulting basin diameter. These basins are comparable with the giant impact

125

basins of Mars created during the heavy bombardment period at around 4 Ga [Frey, 2008].

126

In our models, the impactor was simplified to a spherical body of radius R

imp

with uniform

127

composition while the target was simplified to a two layers spherical body of radius R and an

128

iron core radius of R

core

. The silicate mantle has a thickness of

δm

(See Table 1). We adopt

129

physical properties of silicates (dunite or peridotite) for both the mantle and the impactor to

130

monitor the shock pressure and the particle velocity in a Mars type body. We approximate the

131

thermodynamic response of both the iron and silicate material using the ANEOS equation of

132

state [Thompson and Lauson, 1972, Benz et al., 1989]. To make our models as simple as

133

possible we do not consider here the effects of porosity, thermal softening or low density

134

weakening. However, as a first step towards more realistic models, we investigate the influence

135

of acoustic fluidization and damage. All these effects can be accounted for in iSALE-2D and we

136

will consider each effect in a separate study in near future.

137 138

The early temperature profile of a Mars size body is difficult to constrain because it depends on

139

its accretionary history, on the amount of radiogenic heating and on the mechanisms that led to

140

its core formation [e.g., Senshu et al., 2002]. The uncertainties on the relative importance of

141

these processes as well as the diversity of the processes involved in the core formation lead to a

142

wide range of plausible early thermal states after the full differentiation of Mars. Since we do not

143

consider here the thermal softening during the impact, we assumed the same radially dependent

144

preimpact Martian temperature field in all our simulations. Fig. 1 shows the pre-impact

145

temperature profile as a function of depth. As the pre-impact pressure is governed by the material

146

repartition and as we consider a differentiated Mars, the pre-impact pressure is more

147

straightforward. Fig. 1 also shows the pre-impact hydrostatic pressure used in our models as a

148

function of depth considering a 1700 km thick silicate mantle surrounding a 1700 km radius iron

149

core. We emphasize that the peak pressure shown in our study does not include this hydrostatic

150

pressure. However, the hydrostatic pressure is taken into consideration in calculating the

151

hydrocode models. The peak pressure presents the shock induced pressure increase and is

152

expected to depend on the physical properties of the target, but not on the size of the target as

153

long as the size is large enough to allow shock wave propagates freely without interference with

154

6

reflected waves. A scaling law should reflect the properties of the shock wave propagation in a

155

uniform media. Fig. 2 shows the typical time evolution of the compositional and pressure fields

156

after a 100 km diameter impact with V

imp

=10 km/s. Immediately after the impact, the shockwave

157

propagates downward from the impact site. The shock front reaches the core-mantle boundary in

158

less than 5 minutes while the transient crater is still opening at the surface. It is worth mentioning

159

that the main goal is to derive a scaling law which is useful for numerous impacts during

160

accretion where no information is available about the impact direction, i.e. the impact angle

161

relative to vertical and azimuth relative to north. Moreover, the pressure reduction near the

162

surface due to interference of the direct and reflected waves can easily be accommodated

163

following the procedure by Melosh [1989], which is adapted to spherical surface by Arkani-

164

Hamed [2005], when applying the scaling law to a particular accretion scenario.

165 166

In Fig. 3, we monitor the peak pressure as a function of the distance from the impact site d

167

normalized by the impactor radius R

imp

along the symmetry axis for the case illustrated in Fig. 2.

168

In our iSALE models, the impact-induced pressure fields (as well as temperature and velocity

169

fields) are extracted from a cell-centered Eulerian grid data

.

To validate our models, we have

170

tested different spatial resolutions expressed here in terms of cells per projectile radius (cppr). In

171

Fig. 3a, we represent the peak pressure decrease as a function of depth for cppr values ranging

172

from 5 to 50, showing convergence for cppr values larger than 25. As illustrated in Fig. 3a, the

173

difference between the results with 25 and 50 cppr is less than 10%. This resolution study is in

174

agreement with Pierazzo et al., [2008] who have shown that the iSALE models converge for

175

resolutions of 20 cppr or higher, although a resolution of 10 cppr still provides reasonable results

176

(a resolution of 20 cppr appears to underestimate peak shock pressures by at most 10%). The

177

resolution is 25 cppr or higher in our models.

178

In Pierazzo et al., [1997], the impactor radius ranged between 0.2 and 10 km. In Fig. 3b, we

179

compare our results obtained with R

imp

=10 km, R

imp

=50 km and the results obtained by Pierazzo

180

et al., [1997]. Fig. 3b shows that even with a radius of 50 km, both our results and the results

181

from Pierazzo et al., [1997] are in good agreement, confirming that the impactor size has minor

182

effects on the peak pressure evolution with depth. The small differences between our results with

183

different impactor radii (discussed further in more details) are plausibly the direct consequence

184

of using increasing cppr values with increasing impactor radii. Consequently, we will use the

185

7

normalized distance in all of our figures, as equations of motion should be invariant under

186

rescaling of distance [Melosh, 1989, Pierazzo et al, 1997]. However, to monitor the peak

187

pressure evolution with R

imp

=10 km and to maintain a reasonable computational time, we have

188

used only 10 cppr. This figure confirms that below 10 cppr, the spatial accuracy is insufficient

189

and our results diverge from the results obtained by Pierazzo et al., [1997].

190

Shock pressure and particle velocity scaling laws at low impact velocities:

191 192

A given hydrocode simulation may take on the order of 48 hours to determine a 2D shock

193

pressure and particle velocity distributions in the mantle of our model planet. The impact

194

velocity is about 4 km/s for a protoplanet with a radius of 2860 km and mean density of 3500

195

kg/m

³

, assuming that impacts occur at the escape velocity of the protoplanet. Mars is more likely

196

a runaway planetary embryo formed by accreting small planetesimals and medium size

197

neighboring planetary embryos. This indicates that the accreting bodies had velocities higher

198

than 4 km/s when Mars was growing from 2860 km radius to its present radius of about 3400

199

km. Taking the mean radius of the impacting bodies to be 100 km, which is larger than that of a

200

typical planetesimal, more than 15,000 bodies must have accreted at impact velocities higher

201

than 4 km/s. Calculating the impact induced shock pressure and particle velocity inside the

202

growing Mars would be formidable if hydrocode simulation is adopted for each impact.

203

Because the impact-induced shock pressure P and particle velocity V

p

inside a planetary mantle

204

decease monotonically with distance from the impact side, simple exponential functions have

205

been proposed to estimate peak pressure and particle velocity in the mantle of an impacted body.

206

Solving the shock dynamic equations by a finite difference technique for silicate target and

207

projectile, Ahrens and O’Keefe [1987] showed that pressure distribution in the target displays

208

three regimes: an impedance match regime, Regime I, which extends to 1-3 times the projectile

209

radius into the target where the peak shock pressure is determined by the planar impedance

210

match pressure [McQueen et al., 1970]; a shock pressure decay regime, Regime II, where the

211

pressure decays exponentially as

212

213

P = P

o

(d/R

imp

)

ⁿ

, for d> 1-3 times R

imp,

and n = -1.25 - 0.625 Log(V

imp

), (1)

214

215

8

and the elastic regime, Regime III, where the shock pressure is reduced below the strength of

216

target, the Hugoniot elastic limit, and the shock wave is reduced to an elastic wave with pressure

217

decaying as d

^-3

. In equation (1) d is the distance from the impact site at the surface, R

imp

is the

218

projectile radius, and V

imp

is the impact velocity in km/s. The peak pressure measurements in the

219

nuclear explosions [Perret and Bass, 1975] led Melosh [1989] to propose a scaling low for the

220

particle velocity distribution assuming conservation of momentum of the material behind the

221

shock front which is taken to be a shell of constant thickness. Using several different target

222

materials, and adopting hydrocode simulations Pierazzo et al. [1997] showed that the shock

223

pressure P and particle velocity V

p

decrease slowly in the near field zone, but rapidly in the

224

deeper region,

225

226

P = P

ic

(d

ic

/d)

ⁿ

, n = -1.84 + 2.61 Log(V

imp

), d > d

ic

(2a)

227

V

p

= V

pic

(d

ic

/d)

^m

m = -0.31 + 1.15 Log(V

imp

), d > d

ic

(2b)

228

229

The authors coined an isobaric zone of shock pressure P

ic

and particle velocity V

pic

for the near

230

field of radius d

ic

, about 1.5 R

imp

. Equations 2a and 2b were derived by averaging results from

231

many different materials and impactor sizes. The impact velocities adopted were 10 to 80 km/s

232

and the projectile diameter ranged from 0.4 to 20 km.

233 234

In a log-log plot the peak shock pressure and the corresponding particle velocity are linear

235

functions of distance from the impact site,

236

237

Log P = a + n Log(d/R

imp

) (3a)

238

Log V

p

= c + m Log(d/R

imp

) (3a)

239 240

where a is the logarithm of pressure and c is the logarithm of particle velocity both at R

imp

and n

241

and m are decay factors. All parameters are impact velocity dependent:

242 243

a = α + β Log (V

imp

) (4a)

244

c = γ + Ω Log (V

imp

) (4b)

245 246

9

and

247

n = λ + δ Log (V

imp

) (5a)

248

m = η + ζ Log (V

imp

) (5b)

249 250

Figure 4 shows the peak shock pressure inside the mantle of the model planet we obtained by

251

hydrocode simulation and using a projectile of 100 km diameter at 10 km/s impact velocity.

252

There is actually no isobaric region, rather the peak pressure decays slowly in the near field zone

253

but much rapidly in the deeper parts, similar to the results by Ahrens and O’Keefe [1987].

254

Although the regression lines fitted to the near field and far field are good representatives of the

255

shock pressure distribution, they intersect at a much higher pressure than that of the hydrocode

256

model and overestimate the pressure by as much as 30% in a large region located between the

257

near field and the far field. Therefore, to better approximate the pressure distribution we fit the

258

pressure curve by three lines, representing the near field, mid field and far field regions, which

259

render a much better fit as seen in Figure 4.

260 261

Figure 5a shows the peak shock pressure versus distance from the impact site for impact

262

velocities of 4 to 10 km/s and an impactor of 100 km in diameter. Because of different

263

phenomenon such as excavation, melting, vaporization and intermixing between the target and

264

projectile material, the shock front in near field is more difficult to characterize by our numerical

265

models even for high cppr. Also, as the shock wave is not yet detached from the impactor, it

266

cannot be treated as a single shock wave. Hence, the near field - mid field boundary and the

267

scaling laws for the near field are less accurate than for the two other fields especially for large

268

impact velocities (V

imp

> 7 km/s). In Figure 5, the larger dots show the intersections of the linear

269

regression lines. For example a dot that separates near field from mid field is the intersection of

270

the regression lines fitted to the near field and midfield. The regression lines are determined

271

from fitting to the hydrocode data. Visually, we first divide the hydrocode data of a given model

272

into three separate sections with almost linear trends, and then fit the regression lines to those

273

three trends. Figure 5a shows the hydrocode data, small dots, and the regression lines,

274

demonstrating the tight fitting of the lines to the data. The large dots in the figure show the

275

intersection of the regression lines of the adjacent regions. For example the dot that shows the

276

near-field mid-field boundary is the intersection of the regression lines fitted to the near field and

277

10

mid field hydrocode data. Note that a big dot does not necessarily coincide with the exact

278

hydrocode result, a small dot. As the near field – mid field boundary is relatively less resolved,

279

Figure 5a shows a scatter of the dots separating those two fields and a slight slope change from

280

V

imp

> 7 km/s. In the average the near field – mid field boundary is at ~2.3 R

imp

(~115 km) from

281

the impact site. Figure 5a shows that the depth to the mid field – far field boundary is almost

282

independent of the pressure. It is at about 4.5 R

imp

(~225 km) from the impact site.

283 284

We propose three scaling laws for shock pressure, and three for particle velocity:

285

Log P

i

= a

i

+ n

i

Log(d

i

/ R

imp

), i=1, 2, and 3. (6a)

286

Log V

pi

= c

i

+ m

i

Log(d

i

/ R

imp

), i=1, 2, and 3. (6b)

287

288

For the near field d

i

< 2.3 R

imp

, mid field 2.3 R

imp

<d

i

<4.5 R

imp

, and far field d

i

>4.5 R

imp

. Table

289

2 lists the values of the parameters in the above equations as well as the misfits obtained from

290

our regressions (smaller than 0.001 in all the regressions calculated here). Figure 5a shows the

291

three regression lines fitted to the near field, mid field, and far field of each model for shock

292

pressure, and Figure 5b displays those for particle velocity. The misfits from Table 2 are based

293

on the fixed end points, shown as dots in Fig. 5.

294 295

The close agreement between Pierazzo et al. [1997] model for impact velocity of 10 km/s

296

derived by averaging results of impactors with diameters 0.4 to 20 km, and our result for the

297

same impact velocity but an impactor of 100 km in diameter indicates that the shock pressure

298

distribution is less sensitive to impactor size in a log-log plot of pressure versus distance

299

normalized to the impactor radius. To further investigate this point, we calculate models with

300

impactor sizes of 100 to 400 km in diameter. Figure 6a shows the hydrocode results for impact

301

velocity of 10 km/s using different impactor size. The result obtained for R

imp

=10 km is not

302

included in the figure because of its low cppr value (see Fig. 3b). The curves have almost the

303

same slope in the far field, and small deviations in the mid field and near field.

304 305

Pierazzo et al. [1997] used several different rock types for the solid mantle and concluded that

306

their scaling laws are less sensitive to the rock types. Bearing in mind that dunite and peridotite

307

are probably the most representative rocks for solid mantle, we run a hydrocode model using an

308

11

impactor of 100 km in diameter, impact velocity of 10 km/s, and peridotite as a representative

309

mantle and impactor rock. Figure 6b compares the results for the dunite and peridotite models.

310

They are indeed very similar, especially in the far field region, where the exponential factor n in

311

Equation (3a) is -1.449 for dunite and -1.440 for peridotite. The major differences between the

312

two models arise from the near field zone, hence propagates down to the deeper regions.

313 314

Among the other parameters and phenomenon that may influence the shockwave propagation

315

(porosity, thermal softening…), accounting for the acoustic fluidization is required to accurately

316

simulate the formation of a complex crater [e.g., Melosh, 1979, Bray et al., 2008, Potter et al,

317

2012]. Indeed, acoustic fluidization is invoked to explain the collapse of complex craters by

318

modifying the frictional strength of the damaged target. Figure 6c compares the results obtained

319

for R

imp

=50 km and V

imp

=10 km/s considering acoustic fluidization and an Ivanov damage model

320

which prescribes damage as a function of plastic strain. This figure shows that for the near and

321

mid fields, the results are similar. Pierazzo et al. [1997] did not include acoustic fluidization or

322

damage in their models. Hence, our results without acoustic fluidization or damage model and

323

the results from Pierazzo are in good agreement (Fig. 6c). As soon as the far field is reached,

324

acoustic fluidization and damage tend to reduce the intensity of the shock pressure. This

325

indicates that building more sophisticated models will be necessary in the near future. As the

326

impact heating is mainly localized in the near- and mid-fields, including a damage model or

327

acoustic fluidization should only weakly affect the thermal evolution of a growing protoplanet.

328

However, it is worth mentioning that we are not concerned with the shape of the crater produced

329

by a large impact, rather the main goal of our study is to extend the scaling laws of Pierazzo et

330

al., [1997] to lower impact velocities which are more compatible with accretionary conditions.

331 332

Figure 7 shows the impact velocity dependence of a, n, c, and m. Also included in Figure 7b is

333

the model by Ahrens and O’Keefe [1987] which was derived using impact velocities of 5 km/s

334

and higher. Pierazzo et al. [1997] used impact velocities higher than those considered in the

335

present study, except for their 10 km/s model. Hence, their results are shown in Figure 7b by

336

only one point, asterisk, at the impact velocity of 10 km/s.

337 338

The shock pressure along a non-vertical profile is not supposed to be the same as the one along a

339

12

vertical profile, largely because of the pre-impact lithostatic pressure. As emphasized by

340

Pierazzo et al. [1997, 2008] the shock front in deeper regions appears relatively symmetric

341

around the impact point. Of course, it is not realistic in the case of an oblique impact (not

342

studied here) and for the shallowest angles where the surface significantly affects the shock

343

pressure decay. We have monitored the effect of the shockwave propagation angle θ with values

344

varying between 90° (vertical profile) to 27° (Fig. 8). Similarly to Pierazzo et al [1997], we did

345

not find a significant angle dependence on our results especially when θ is ranging between 90°

346

and 45°. For smaller values of θ, the surface effects appear to modify the shockwave propagation

347

by reducing its intensity (Fig. 8 a). Except in the mid field, where the n value decreases from -0.6

348

to -1.31, and in the far field, where the a value decreases from 2.54 to 1.93, the coefficients a and

349

n from our scaling laws do not change significantly with the angle (Fig. 8 b). This is particularly

350

true in the near field where most of the impact heating occurs.

351

352

Scaling laws have been used by many investigators [e.g., Senshu et al., 2002; Tonks and Melosh,

353

1992, 1993; Watters et al, 2009; Roberts et al., 2009; Arkani-Hamed and Olson, 2010a, 2010b;

354

Arkani-Hamed and Ghods, 2011], mainly because they require a much smaller computer and

355

much less computer time and the difference between a hydrocode model and a corresponding

356

scaling model is minute. For example, Figure 9 shows the 2D distribution of the peak shock

357

pressure determined for an impactor of 100 km in diameter and an impact velocity of 10 km/s

358

calculated using our scaling laws in near field, mid field, and far field, and the parameter values

359

from Table 2. The grid spacing is 2 km in radial direction and 0.03 degrees in the colatitude

360

direction. The entire computer time in a PC, CPU: 2.4 GHz, was only 16 seconds, which also

361

calculated the 2D distribution of shock-induced temperature increase using Watters et al.’s

362

[2009] foundering shock heating model. The computer time is substantially shorter than 48

363

hours taken by our corresponding hydrocode model using a CPU: 2.9 GHz laptop. This shows

364

that it is feasible to determine impact heating during the accretion of a terrestrial planet using

365

scaling laws, whereas it is almost impossible to adopt hydrocode simulations for that purpose.

366 367

During the decompression of shocked material much of the internal energy of the shock state is

368

converted into heat [O’Keefe and Ahrens, 1977]. Using thermodynamic relations, the waste heat

369

used to heat up the impacted material can be estimated [Gault and Heitowit, 1963; Watters et al.,

370

13

2009] and the corresponding temperature increase ΔT can be obtained. Hence, on the basis of

371

our scaling laws it is possible to estimate the temperature increase as a function of depth below

372

the impact site for impact velocities compatible with the accretionary conditions of terrestrial

373

protoplanets. These scaling laws can easily be implemented in a multi-impact approach [e.g.

374

Senshu et al, 2003, Monteux et al., 2014] to monitor the temperature evolution inside a growing

375

protoplanet whereas it is not yet possible to adopt hydrocode simulations for that purpose. For

376

example, included in Figure 9 is the impact induced temperature increase corresponding to the

377

shock pressure shown in the figure. The temperature increase is determined on the basis of

378

foundering model of Watters et al. [2009] using constant values for the acoustic velocity C (6600

379

m/s) and the parameter S (0.86) in their expressions:

380 381

∆𝑇(𝑃)= _!!^!

! ! 1−𝑓^!! −(𝐶/𝑆)² 𝑓−ln𝑓−1

(7)

382

𝑓 𝑃 =−^!

! 1− ^!!

! +1

!!

(8)

383

𝛽= ^𝐶_!!^2!!

(9)

384 385

with P the shock-increased pressure and ρ

0

the density prior to shock compression (see Tab. 1 for

386

values).

387 388

Due to small size the impactor is not capable of increasing the lower mantle temperature of the

389

model planet significantly, and only minor impact heating of the core has occurred. The thermal

390

evolution model has to be combined to a topographical evolution model to account for the

391

growth of the protoplanet as in Monteux et al., [2014]. In these models, the impact angle

392

(considered here as vertical) probably plays a key role because it influences both the morphology

393

of the impact heating and the shape of the post-impact topography. A more elaborated scaling

394

laws built upon 3D hydrocode models will be developed for that purpose in the near future.

395 396

It is worth emphasizing that our scaling laws, like those of others [Ahrens and O’Keefe, 1987;

397

Pierazzo et al., 1997; Mitani, 2003], are derived from a few hydrocode models. Figure 10 shows

398

the profiles of the pressure along the axis of symmetry for comparison. The differences between

399

14

the hydrocode model and the scaling law are small for the most part, but the exact scaling law

400

differs by ~10 GPa for d/R

imp

= 2-3. This difference arises from the difficulty of correctly

401

describe the near field as previously mentioned. Note that the interpolated model is in much

402

better agreement with the hydrocode model.

403 404

A linear relationship has been proposed between the shock wave velocity V

s

and particle velocity

405

V

p

on the basis of laboratory measurements [McQueen, 1967; Trunin, 2001]

406 407

V

s

= C + S V

p

(10)

408 409

where C is the acoustic velocity and S is a constant parameter. We estimate the acoustic velocity

410

in the mantle of the model planet on the basis of our hydrocode models (Figure 5a, 5b) using

411

Equation (10) and the Hugoniot equation

412

413

P = ρ V

p

V

s

(11)

414 415

where ρ (=3320 kg/m

³

) is the pre-shock density. Figure 11 shows the variations of C with depth

416

for models with impact velocities of 4 to 10 km/s and an impactor of 100 km diameter, using

417

S=1.2 which is within the values proposed by the authors for dunite [e.g., Trunin, 2001].

418

Adopting S=0.86 [McQueen, 1967] does not change the results significantly, especially in the far

419

field, where the acoustic velocity is less sensitive to particle velocity and linearly increases with

420

depth. However C shows particle velocity dependence in the mid field and near field.

421 422

Conclusions:

423 424

We have modeled the shock pressure and particle velocity distributions in the mantle of a Mars

425

size planet using hydrocode simulations (iSALE-2D) for impact velocities of 4 to 10 km/s and

426

projectile diameters ranging from 100 to 400 km. We have extended Pierazzo et al.’s [1997]

427

scaling laws to low impact velocities and also considered large impactor radii occurring during

428

the formation of terrestrial planets. We propose three distinct regions in the mantle: a near field

429

region, which extends to 1-3 times the projectile radius into the target, where the peak shock

430

15

pressure and particle velocity decay very slowly with increasing distance, a mid field region,

431

which extends to ~ 4.5 times the impactor radius, where the pressure and particle velocity decay

432

exponentially but moderately, and a more distant far field region where the pressure and particle

433

velocity decay strongly with distance. The mid field – far field boundary is well constrained,

434

whereas that of the near field - mid field is a relatively broad transition zone for the impact

435

velocities examined.

436 437

Acknowledgements: This research was supported by Agence Nationale de la Recherche

438

(Oxydeep decision No. ANR-13-BS06-0008) to JM, and by Natural Sciences and Engineering

439

Research Council (NSERC) of Canada to JAH. We gratefully acknowledge the developers of

440

iSALE (www.isale-code.de), particularly the help we have received from Gareth S. Collins. We

441

are also grateful to the two reviewers for very helpful suggestions.

442 443

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444

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445

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572

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573

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574 575 576 577 578 579 580

20

Table List:

581 582

Table 1. Typical parameter values for numerical hydrocode models

583

584

Target radius R 3400 km

Target core radius R

core

1700 km

Silicate mantle thickness

δm

1700 km

Impactor radius R

imp

50-200 km

Impact velocity V

imp

4-10 km/s

Mantle properties (Silicates)

Initial density

ρm

3314 kg/m

³

Equation of state type ANEOS

Poisson

Strength Model

(iSALE parameters) Acoustic Fluidization Model (iSALE parameters) Damage Model

(iSALE parameters) Thermal softening and porosity models

0.25 Rock

(Y

i0

=10 MPa, µ

i

=1.2,Y

im

=3.5 GPa) Block

(t

off

=16 s, c

vib

=0.1 m/s, vib

max

=200 m/s) Ivanov

(ε

fb

=10

^-4

, B=10

^-11

, p

c

=3x10

⁸

Pa) None

Core properties (Iron)

Initial density

ρc

7840 kg/m

³

Equation of state type ANEOS

Poisson

Strength Model

(iSALE parameters) Acoustic Fluidization Model (iSALE parameters) Damage, thermal softening and porosity models

0.3

Von Mises (Y

0

=100 MPa) Block

(t

off

=16 s, c

vib

=0.1 m/s, vib

max

=200 m/s)

None

21 585

586 587 588

Table 2. Parameters of the peak shock pressure distribution and the corresponding particle

589

velocity in the mantle of the Mars size model planet. The pressure is expressed as:

590 591

Log(P) = a + n Log(d/R

imp

)

592

593

and the particle velocity as:

594 595

Log(V

p

) = c + m Log(d/R

imp

)

596

597

where the pressure P is in GPa, the particle velocity V

p

is in km/s, d is the distance from the

598

impact site at the surface, and R

imp

is the impactor radius. a and c are the logarithm of pressure

599

and particle velocity at the distance R

imp

from the impact site, and n and m are the decay

600

exponents of pressure and particle velocity with distance from the impact site. a, c, n and m are

601

impact velocity dependent:

602 603

a = α + β Log(V

imp

)

604

n = λ + δ Log(V

imp

)

605

c = γ + Ω Log(V

imp

)

606

m = η + ζ Log(V

imp

)

607

608

A misfit value is obtained by calculating the standard deviation of a line fitted to the hydrocode

609

data within a given region: 𝜖

= 1/N ^!_!(𝑌_data−𝑌_regression)^!

, where N is the total number of

610

points, Y

data

is the hydrocode result and Y

regression

denotes the value obtained by the linear

611

regression. The zero misfit implies that the regression line is fitted to only 2 points, hence an

612

exact fitting.

613 614 615

Near Field

616

V

imp

a n misfit c m misfit

617

(km/s)

618

4 1.1717 -0.4530 1.074E-07 0.1276 -1.1132 3.071E-08

619

5 1.3963 -0.6296 4.616E-03 0.2901 -1.0437 6.6837E-04

620

6 1.5137 -0.4713 8.429E-08 0.3722 -0.8573 0.000

621

7 1.6527 -0.3237 5.960E-08 0.3315 -0.4286 1.490E-08

622

8 1.8093 -0.3302 5.960E-08 0.5736 -0.6567 0.000

623

9 1.8853 -0.1228 8.429E-08 0.6817 -0.2156 4.214E-08

624

10 1.9072 -0.1364 1.332E-07 0.7354 -0.2622 8.411E-03

625

α = 0.040, β = 1.914, λ = -1.214, δ = 1.058 626

γ = -0.795, Ω = 1.502 η = -2.602, ζ = 2.368 627

628 629

22

Mid Field

630

V

imp

a n misfit c m misfit

631

(km/s)

632

4 1.3714 -0.9459 2.0576E-03 0.0917 -1.0211 3.8861E-04

633

5 1.5978 -0.9995 6.5599E-04 0.2911 -1.0402 7.3517E-04

634

6 1.6367 -0.8038 4.7266E-03 0.4143 -0.9942 8.0339E-04

635

7 1.8821 -0.9792 2.8677E-03 0.5141 -0.9563 1.8048E-03

636

8 1.9735 -0.8576 3.5979E-03 0.6750 -0.9888 6.5566E-05

637

9 2.0060 -0.7072 6.2120E-03 0.8611 -1.1259 2.0130E-03

638

10 2.0224 -0.6059 8.2497E-03 0.9317 -1.1190 1.3516E-03

639

α = 0.346, β = 1.736, λ = -1.469, δ = 0.768 640

γ = -1.206, Ω = 2.114, η = -0.864, ζ = -0.208 641

642

643

Far Field

644

V

imp

a n misfit c m misfit

645

(km/s)

646

4 1.5136 -1.1453 6.4139E-04 0.2397 -1.2158 1.1549E-03

647

5 1.7356 -1.1640 5.0752E-04 0.4635 -1.2389 1.0468E-03

648

6 1.9107 -1.1864 6.9751E-04 0.6248 -1.2531 1.2862E-03

649

7 2.0602 -1.2182 6.0862E-04 0.7628 -1.2783 1.3663E-03

650

8 2.2186 -1.2816 3.3932E-04 0.8936 -1.3220 1.0471E-03

651

9 2.4057 -1.3818 1.1730E-03 1.0620 -1.4091 8.7868E-04

652

10 2.5440 -1.4492 1.5957E-03 1.1887 -1.4687 1.3772E-03

653

α = -0.056, β = 2.558, λ = -0.647, δ = -0.744 654

γ = -1.177, Ω = 2.333, η = -0.818, ζ = -0.600 655

656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675

23 676

677 678 679 680

681

Figure 1

682

Figure 1: Pre-impact temperature (left) and lithostatic pressure (right) as a function of depth.

683

The dashed lines illustrate the core-mantle boundary.

684 685 686

24 687

Figure 2: A close up view of the material repartition (left column) and total pressure (right

688

column) as functions of time (from top to bottom) in the model planet (for V

imp

=10 km/s and

689

D

imp

=100 km). In this model, the grid resolution is 2 km in all directions. The silicate mantle

690

and the impactor are made of dunite.

691 692 693 694

t = 4 min t = 2 min

t = 30 s t = 0

Pressure (GPa)

0 30

Material

Core Mantle

25 695

696 697 698 699

700

Figure 3a Figure 3b

701 702 703

Figure 3: Peak pressure decrease as a function of depth normalized by the radius of the impactor

704

for the impact velocity of 10 km/s. The silicate mantle as well as the impactor are made of

705

dunite. 3a: Influence of the spatial resolution. Here we only consider the case with R

imp

=50 km.

706

The results from our hydrocode models are shown by colored curves with a spatial resolution

707

ranging from 5 to 50 cppr. 3b: Comparison of our results with R

imp

=50 km (red curve, 25 cppr)

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and R

imp

=10 km (green curve, 10 cppr) with the results from a similar model of Pierazzo et al.,

709

(1997) (black squares).

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26 724

725 726

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Figure 4. Shock pressure versus normalized distance from the impact site at the surface

728

produced by a 100 km diameter impactor with an impact velocity of 10 km/s. The dashed curve

729

represents the hydrocode model, while the straight lines are fitted to three different parts of the

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hydrocode model.

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27 733

734

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Figure 5a Figure 5b

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Figure 5a. Shock pressure versus normalized distance from the impact site at the surface for an

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impactor of 100 km diameter and impact velocities ranging from 4 to 10 km/s. The numbers on

739

the curves are the impact velocities. The hydrocode results are presented by dots, while the

740

regression lines to the near field, mid field and far field regions are straight lines. The larger dots

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show the intersections of the linear regression lines. For example a dot that separates near field

742

from mid field is the intersection of the regression lines fitted to the near field and mid field data.

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5b. shows the corresponding particle velocity.

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28 746

747

Figure 6a Figure 6b

748 749 750

751

Figure 6c

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Figure 6a. shows the hydrocode results for impactors of 50 to 400 km diameter and impact

754

velocity of 10 km/s. 6b. compares the hydrocode results using dunite and peridotite as mantle

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rock types, for an impactor of 100 km diameter and impact velocity of 10 km/s. 6c illustrates the

756

shock pressure as a function of d/R

imp

with (red curve) and without (black curve) acoustic

757

fluidization. The green curve represents the results considering an Ivanov damage model. (For

758

comparison, the black squares represent the results from Pierazzo et al., (1997], which has not

759

considered acoustic fluidization).

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29 763

764

765

Figure 7a Figure 7b

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Figure 7c Figure 7d

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Figure 7. Dependence of regression parameters a, n, c, and m from Eq. 4 and 5 on the impact

771

velocity. Dots are based on hydrocode models and lines are regression fits, see Table 2

772

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